Question

How do you find an equation that describes the sequence `16, 17, 18, 19,...` and find the 23rd term?

Answer 1

`a_n = n + 15`
`a_23 = 38`

Explanation:

To find an equation, we should use the arithmetic sequence formula:

`a_n = a_1 + (n-1)d`

When using this formula, you need to find the values for `d` and `a_1`.

1. Finding `d` (common difference)

We are given the sequence `16, 17, 18, 19, ...`. Using this information, we can find the common difference (`d`), which is another way of saying the difference between any two consecutive numbers in the arithmetic sequence. You could find `d` in a number of ways:

`17 - 16 = 1`
`18 - 17 = 1`

etc.

But whatever way you choose to find it, you should get that `d = 1`.

2. Finding `a_1`

`a_1` is the first term of the sequence. In our case, `a_1` = 16.

3. Plug into the formula.

`a_n = 16 + (n-1)(1)`

Now distribute `1` to `(n-1)`:

`a_n = 16 + n - 1`
`a_n = n + 15`

That's your equation!

Now plug in 23 for `n`:

`a_23 = 23 + 15`
`a_23 = 38`

Answer 2

`a_n=n+15" and " a_(23)=38`

Explanation:

`" this is an arithmetic sequence"`

`"the nth term is " a_n=a+(n-1)d`

`"where " a " is the first term and " d" the common difference"`

`"here " a=16" and " d=1`

`rArra_n=16+n-1=n+15`

`rArra_(23)=23+15=38`